Junio 2012, volumen 35, no. 1, pp. 67 a 76
Estimation of Reliability in Multicomponent Stress-strength Based on Generalized Exponential
Distribution
Estimación de confiabilidad en la resistencia al estrés de multicomponentes basado en la distribución exponencial generalizada
Gadde Srinivasa Raoa
Department of Statistics, School of Mathematical and Computer Sciences, Dilla University, Dilla, Ethiopia
Abstract
A multicomponent system ofkcomponents having strengths followingk- independently and identically distributed random variables X1, X2, . . . , Xk
and each component experiencing a random stress Y is considered. The system is regarded as alive only if at least sout of k(s < k) strengths ex- ceed the stress. The reliability of such a system is obtained when strength and stress variates are given by generalized exponential distribution with different shape parameters. The reliability is estimated using ML method of estimation in samples drawn from strength and stress distributions. The reliability estimators are compared asymptotically. The small sample com- parison of the reliability estimates is made through Monte Carlo simulation.
Using real data sets we illustrate the procedure.
Key words:Asymptotic confidence interval, Maximum likelihood estima- tion, Reliability, Stress-strength model.
Resumen
Se considera un sistema de k multicomponentes que tiene resistencias que se distribuyen como k variables aleatorias independientes e idéntica- mente distribuidasX1, X2, . . . , Xky cada componente experimenta un estrés aleatorio Y. El sistema se considera como vivo si y solo si por lo menos s dek(s < k) resistencias exceden el estrés. La confiabilidad de este sistema se obtiene cuando las resistencias y el estrés se distribuyen como una dis- tribución exponencial generalizada con diferentes parámetros de forma. La confiabilidad es estimada usando el método ML de estimación en muestras extraídas tanto para distribuciones de resistencia como de estrés. Los esti- madores de confiabilidad son comparados asintóticamente. La comparación
aProfessor. E-mail: [email protected]
para muestras pequeñas de los estimadores de confiabilidad se hace a través de simulaciones Monte Carlo. El procedimiento también se ilustra mediante una aplicación con datos reales.
Palabras clave:confiabilidad, estimación máximo verosímil, intervalos de confianza asintóticos, modelo de resistencia-estrés.
1. Introduction
The two-parameter generalized exponential distribution (GE) has been intro- duced and studied quite extensively by Gupta & Kundu (1999, 2001, 2002). The two-parameter GE distribution is an alternative to the well known two-parameter gamma, two-parameter Weibull or two parameter log-normal distributions. The two-parameter GE distribution has the following density function and the distri- bution function, respectively
f(x;α, λ) =αλe−xλ(1−e−xλ)α−1; forx >0 (1)
F(x;α, λ) = (1−e−xλ)α−1; forx >0 (2) Here αandλare the shape and scale parameters, respectively. Now onwards GE distribution with the shape parameterαand scale parameterλwill be denoted by GE(α,λ).
The purpose of this paper is to study the reliability in a multicomponent stress-strength based on X, Y being two independent random variables, where X∼GE(α, λ) andY ∼GE(β,λ).
Let the random samplesY, X1, X2, . . . , Xkbeing independent,G(y)be the con- tinuous distribution function ofY andF(x)be the common continuous distribution function of X1, X2, . . . , Xk. The reliability in a multicomponent stress-strength model developed by Bhattacharyya & Johnson (1974) is given by
Rs,k=P[at leastsof theX1, X2, . . . , Xk exceed Y]
=
k
X
i=s
k i
Z ∞
−∞
[1−F(y)]i [F(y)](k−i)dG(y) (3)
WhereX1, X2, . . . , Xkare independently identically distributed (iid) with com- mon distribution functionF(x), this system is subjected to common random stress Y. The probability in (3) is called reliability in a multicomponent stress-strength model (Bhattacharyya & Johnson 1974). The survival probability of a single com- ponent stress-strength version has been considered by several authors assuming various lifetime distributions for the stress-strength random variates, e.g. Enis
& Geisser (1971), Downtown (1973), Awad & Gharraf (1986), McCool (1991), Nandi & Aich (1994), Surles & Padgett (1998), Raqab & Kundu (2005), Kundu &
Gupta (2005), Kundu & Gupta (2006), Raqab, Modi & Kundu (2008), Kundu &
Raqab (2009). The reliability in a multicomponent stress-strength was developed
by Bhattacharyya & Johnson (1974), Pandey & Uddin (1985), and the references therein cover the study of estimating in many standard distributions assigned to one or both stress, strength variates. Recently, Rao & Kantam (2010) studied estimation of reliability in multicomponent stress-strength for the log-logistic dis- tribution.
Suppose that a system, withkidentical components, functions ifs(1≤s≤k) or more of the components simultaneously operate. In this operating environment, the system is subjected to a stressY which is a random variable with distribution functionG(.). The strengths of the components, that is the minimum stress to cause failure, are independent and identically distributed random variables with distribution functionF(.). Then, the system reliability, which is the probability that the system does not fail, is the function Rs,k given in (3). The estimation of the survival probability in a multicomponent stress-strength system when the stress follows a two-parameter GE distribution has not received much attention in the literature. Therefore, an attempt is made here to study the estimation of reliability in multicomponent stress-strength model with reference to the two- parameter GE probability distribution. In Section 2, we derive the expression for Rs,k and develop a procedure for estimating it. More specifically, we obtain the maximum likelihood estimates of the parameters. The Maximum Likelihood Estimators (MLEs) are employed to obtain the asymptotic distribution and confi- dence intervals forRs,k. The small sample comparisons are made through Monte Carlo simulations in Section 3. Also, using real data, we illustrate the estimation process. Finally, some conclusion and comments are provided in Section 4.
2. Maximum Likelihood Estimator of R
s,kLet X ∼ GE(α, λ) and Y ∼ GE(β, λ) with unknown shape parameters α and β and common scale parameter λ, where X and Y are independently dis- tributed. The reliability in multicomponent stress-strength for two-parameter GE distribution using (3) is
Rs,k=
k
X
i=s
k i
Z ∞ 0
[1−(1−e−yλ)α]i[(1−e−yλ)α](k−i)βλe−yλ(1−e−yλ)β−1dy
=
k
X
i=s
k i
Z 1 0
[1−tν]i [tν](k−i)dt, where t= (1−e−yλ)β and ν =α β
= 1 ν
k
X
i=s
k i
Z 1 0
[1−z]i [z](k−i+1ν−1)dz if z=tν
= 1 ν
k
X
i=s
β(k−i+1 ν, i+ 1)
After the simplification we get
Rs,k= 1 ν
k
X
i=s
k!
(k−i)!
i
Y
j=0
(k+1 ν −j)
−1
, sincekandiare integers (4)
The probability in (4) is called reliability in a multicomponent stress-strength model. Ifαandβ are not known, it is necessary to estimateαandβ to estimate Rs,k. In this paper we estimate α and β by the ML method. Once MLEs are obtained thenRs,kcan be computed using equation (4).
Let X1, X2, . . . , Xn; Y1, Y2, . . . , Ym be two ordered random samples of sizen, m, respectively, on strength, stress variates following a GE distribution with shape parametersαandβand a common scale parameterλ. The log-likelihood function of the observed sample is
L(α, β, λ) = (m+n) lnλ+nlnα+mlnβ−λ
n
X
i=1
xi−
m
X
j=1
yj
+
(α−1)
n
X
i=1
ln(1−e−xiλ) + (β−1)
m
X
j=1
ln(1−e−yjλ)
(5)
The MLEs ofα, β andλ, sayα,b βbandbλ, respectively, can be obtained as the solution of
αb= −n
n
P
i=1
ln(1−e−xiλ)
(6)
βb= −m
m
P
j=1
ln(1−e−yjλ)
(7)
g(λ) = 0⇒m+n
λ −
n
n
P
i=1
xi e−xiλ 1−e−xiλ n
P
k=1
ln(1−e−xkλ)
− m
m
P
j=1
yj e−yj λ 1−e−yj λ m
P
k=1
ln(1−e−ykλ)
−
n
X
i=1
xi 1−e−xiλ −
m
X
j=1
yj 1−e−yjλ
(8)
Therefore, bλ is a simple iterative solution of the non-linear equationg(λ) = 0. Once we obtain λ;b αb and βb can be obtained from (6) and (7), respectively.
Therefore, the MLE ofRs,kbecomes
Rbs,k= 1 νb
k
X
i=s
k!
(k−i)!
i
Y
j=0
(k+1 νb−j)
−1
, where bν= αb
βb (9)
To obtain the asymptotic confidence interval forRs,k, we proceed as below:
The asymptotic variance of the MLE is given by V(α) =b
E(−∂2L
∂α2)
= α2
n and V(bβ) =
E(−∂2L
∂β2)
= β2
n (10)
The asymptotic variance (AV) of an estimate ofRs,kwhich is a function of two independent statisticsαb andβbis given by Rao (1973).
AV(Rbs,k) =V(α)b ∂Rs,k
∂α 2
+V(β)b ∂Rs,k
∂β 2
(11) From the asymptotic optimum properties of MLEs (Kendall & Stuart 1979) and of linear unbiased estimators (David 1981), we know that MLEs are asymptotically equally efficient having the Cramer-Rao lower bound as their asymptotic variance, as given in (10). Thus, from equation (11), the asymptotic variance ofRbs,k can be obtained. To avoid the difficulty of the derivation of the Rs,k, we obtain the derivatives ofRs,kfor(s, k)=(1,3) and (2,4) separately and they are given by
∂R1,3
∂α = 3
β(3νb+ 1)2 and ∂R1,3
∂β = −3νb β(3νb+ 1)2
∂R2,4
∂α = 12bν(7bν+ 2)
β[(3bν+ 1)(4bν+ 1)]2 and ∂R2,4
∂β = −12bν2(7νb+ 2) β[(3νb+ 1)(4νb+ 1)]2 ThusAV(Rb1,3) =(39νb2
bν+1)4 1 n +m1
AV(Rb2,4) = 144νb4(7bν+ 2)2 [(3νb+ 1)(4bν+ 1)]4
1 n + 1
m
as n → ∞, m → ∞,Rbs,k−Rs,k
AV(Rds,k)
−→d N(0,1) and the asymptotic confidence 95%
confidence interval forRs,k is given by Rbs,k±1.96
q
AV(Rbs,k)
The asymptotic confidence 95% confidence interval forR1,3 is given by
Rb1,3±1.96 3bν (3bν+ 1)2
s1 n+ 1
m
, where νb=αb βb
The asymptotic confidence 95% confidence interval forR2,4 is given by
Rb2,4±1.96 12bν2(7νb+ 2) [(3νb+ 1)(4νb+ 1)]2
s1 n+ 1
m
, where νb=αb βb
3. Simulation Study and Data Analysis
3.1. Simulation Study
In this subsection we present some results based on Monte Carlo simulations to compare the performance of theRs,k using different sample sizes. 3,000 ran- dom samples of size 10(5)35 each from stress population, strength population are generated for (α, β) = (3.0,1.5), (2.5,1.5), (2.0,1.5), (1.5,1.5), (1.5,2.0),(1.5,2.5) and (1.5,3.0) in line with Bhattacharyya & Johnson (1974). The MLE of scale parameterλis estimated by the iterative method, and the usingλthe shape pa- rametersαandβ are estimated from (6) and (7). These ML estimators ofαand β are then substituted in ν to get the reliability in a multicomponent reliability for(s, k) = (1,3),(2,4). The average bias and average mean square error (MSE) of the reliability estimates over the 3000 replications are given in Tables 1 and 2.
Average confidence length and coverage probability of the simulated 95% confi- dence intervals ofRs,kare given in Tables 3 and 4. The true values of reliability in multicomponent stress-strength with the given combinations for(s, k) = (1,3) are 0.857, 0.833, 0.800, 0.750, 0.692, 0.643, 0.600, and for(s, k) = (2,4)are 0.762, 0.725, 0.674, 0.600, 0.519, 0.454, and 0.400. Thus, the true value of reliability in multicomponent stress-strength model decreases as β increases for a fixed α whereas reliability in multicomponent stress-strength increases as increases for a fixedβ in both the cases (s, k). Therefore, the true value of reliability decreases as ν decreases, and vice versa. The average bias and average MSE decrease as sample size increases for both methods of estimation in reliability. Also the bias is negative in both situations of(s, k). It verifies the consistency property of the MLE of Rs,k. Whereas, among the parameters the absolute bias and MSE de- crease as α increases for a fixed β in both cases of (s, k) and the absolute bias and MSE increase as β increases for a fixed α in both the cases of (s, k). The length of the confidence interval also decreases as the sample size increases. The coverage probability is close to the nominal value in all cases but slightly less than 0.95. Overall, the performance of the confidence interval is quite good for all com- binations of parameters. Whereas, among the parameters we observed the same phenomenon for average length and average coverage probability that we observed in the case of average bias and MSE.
3.2. Data Analysis
In this subsection we analyze two real data sets and demonstrate how the pro- posed methods can be used in practice. The first data set is reported by Lawless (1982) and the second one is given by Linhardt & Zucchini (1986). Both are ana- lyzed and fitted for various lifetime distributions. We fit the generalized exponen- tial distribution to the two data sets separately. The first data set (Lawless 1982, p. 228) presented here arose in tests on endurance of deep groove ball bearings.
The data presented are the number of million revolutions before failure for each of the 23 ball bearings in the life test, and they are: 17.88, 28.92, 33.00, 41.52, 42.12, 45.60, 48.80, 51.84, 51.96, 54.12, 55.56, 67.80, 68.64, 68.64, 68.88, 84.12, 93.12,
98.64, 105.12, 105.84, 127.92, 128.04, and 173.40. Gupta & Kundu (2001) studied the validity of the model and they compute the Kolmogorov-Smirnov (KS) distance between the empirical distribution function and the fitted distribution functions of generalized exponential distribution which is 0.1058 with a correspondingp-value of 0.9592.
Table 1: Average bias of the simulated estimates of Rs,k. (α,β)
(s, k) (n, m) (3.0,1.5) (2.5,1.5) (2.0,1.5) (1.5,1.5) (1.5,2.0) (1.5,2.5) (1.5,3.0) (10,10) −0.0029 −0.0047 −0.0072 −0.0109 −0.0150 −0.0183 −0.0207 (15,15) −0.0021 −0.0042 −0.0058 −0.0081 −0.0105 −0.0123 −0.0137 (1,3) (20,20) −0.0018 −0.0027 −0.0039 −0.0058 −0.0079 −0.0096 −0.0109 (25,25) −0.0012 −0.0020 −0.0030 −0.0046 −0.0064 −0.0078 −0.0089 (30,30) −0.0011 −0.0019 −0.0028 −0.0041 −0.0055 −0.0066 −0.0075 (35,35) −0.0002 −0.0006 −0.0012 −0.0021 −0.0031 −0.0040 −0.0047 (10,10) −0.0029 −0.0039 −0.0063 −0.0092 −0.0116 −0.0128 −0.0131 (15,15) −0.0022 −0.0034 −0.0059 −0.0075 −0.0087 −0.0092 −0.0091 (2,4) (20,20) −0.0017 −0.0027 −0.0040 −0.0056 −0.0070 −0.0077 −0.0080 (25,25) −0.0010 −0.0019 −0.0030 −0.0044 −0.0056 −0.0063 −0.0065 (30,30) −0.0009 −0.0011 −0.0030 −0.0041 −0.0051 −0.0057 −0.0059 (35,35) −0.0003 −0.0002 −0.0008 −0.0016 −0.0023 −0.0027 −0.0029
Table 2: Average MSE of the simulated estimates ofRs,k. (α,β)
(s, k) (n, m) (3.0,1.5) (2.5,1.5) (2.0,1.5) (1.5,1.5) (1.5,2.0) (1.5,2.5) (1.5,3.0) (10,10) 0.0041 0.0052 0.0068 0.0092 0.0119 0.0139 0.0153 (15,15) 0.0026 0.0033 0.0043 0.0058 0.0075 0.0087 0.0096 (1,3) (20,20) 0.0017 0.0022 0.0029 0.0040 0.0052 0.0061 0.0068 (25,25) 0.0014 0.0018 0.0024 0.0032 0.0042 0.0050 0.0055 (30,30) 0.0011 0.0014 0.0018 0.0025 0.0032 0.0038 0.0043 (35,35) 0.0009 0.0011 0.0015 0.0021 0.0027 0.0032 0.0036 (10,10) 0.0096 0.0115 0.0141 0.0171 0.0193 0.0199 0.0196 (15,15) 0.0062 0.0075 0.0091 0.0111 0.0125 0.0130 0.0128 (2,4) (20,20) 0.0042 0.0051 0.0063 0.0078 0.0090 0.0094 0.0094 (25,25) 0.0035 0.0043 0.0052 0.0065 0.0074 0.0078 0.0078 (30,30) 0.0028 0.0033 0.0041 0.0050 0.0058 0.0060 0.0060 (35,35) 0.0022 0.0027 0.0034 0.0042 0.0049 0.0052 0.0052
The second data set (from Linhardt & Zucchini 1986, p. 69) represents the failure times of the air conditioning system of an airplane (in hours): 23, 261, 87, 7, 120, 14, 62, 47, 225, 71, 246, 21, 42, 20, 5, 12, 120, 11, 3, 14, 71, 11, 14, 11, 16, 90, 1, 16, 52, 95. Gupta & Kundu (2003) studied the validity of the generalized exponential distribution and they compute the Kolmogorov-Smirnov (KS) distance between the empirical distribution function and the fitted distribution functions which is 0.1744 with a correspondingp-value 0.2926. Therefore, it is clear that the generalized exponential model fits quite well to both the data sets.
We use the iterative procedure to obtain the MLE ofλusing (8), and MLEs of αandβ are obtained by substituting MLE ofλin (6) and (7). The final estimates arebλ= 2.80609,αb= 1.00667 andβb= 0.02098. Based on the estimates ofαand
β, the MLE of Rs,k becomes Rb1,3 = 0.893191 and Rb2,4 = 0.819677. The 95%
confidence intervals for R1,3 become (0.841368, 0.945014) and for R2,4 become (0.735472, 0.903882).
Table 3: Average confidence length of the simulated 95% confidence intervals ofRs,k. (α,β)
(s, k) (n, m) (3.0,1.5) (2.5,1.5) (2.0,1.5) (1.5,1.5) (1.5,2.0) (1.5,2.5) (1.5,3.0) (10,10) 0.2112 0.2399 0.2762 0.3221 0.3627 0.3873 0.4012 (15,15) 0.1747 0.1981 0.2279 0.2659 0.3000 0.3212 0.3337 (1,3) (20,20) 0.1512 0.1716 0.1977 0.2311 0.2614 0.2804 0.2918 (25,25) 0.1351 0.1534 0.1768 0.2069 0.2342 0.2515 0.2619 (30,30) 0.1238 0.1404 0.1618 0.1893 0.2145 0.2304 0.2401 (35,35) 0.1140 0.1295 0.1492 0.1748 0.1982 0.2132 0.2224 (10,10) 0.3267 0.3628 0.4045 0.4485 0.4744 0.4782 0.4697 (15,15) 0.2721 0.3020 0.3368 0.3742 0.3973 0.4020 0.3962 (2,4) (20,20) 0.2366 0.2630 0.2939 0.3274 0.3486 0.3533 0.3486 (25,25) 0.2119 0.2356 0.2635 0.2939 0.3134 0.3180 0.3141 (30,30) 0.1943 0.2161 0.2416 0.2697 0.2878 0.2923 0.2890 (35,35) 0.1794 0.1996 0.2234 0.2497 0.2669 0.2716 0.2688
Table 4: Average coverage probability of the simulated 95% confidence intervals ofRs,k. (α,β)
(s, k) (n, m) (3.0,1.5) (2.5,1.5) (2.0,1.5) (1.5,1.5) (1.5,2.0) (1.5,2.5) (1.5,3.0) (10,10) 0.9230 0.9247 0.9277 0.9220 0.9140 0.9070 0.9053 (15,15) 0.9327 0.9330 0.9357 0.9323 0.9303 0.9280 0.9243 (1,3) (20,20) 0.9373 0.9387 0.9397 0.9400 0.9360 0.9293 0.9243 (25,25) 0.9287 0.9323 0.9347 0.9360 0.9340 0.9293 0.9247 (30,30) 0.9347 0.9360 0.9393 0.9403 0.9420 0.9427 0.9363 (35,35) 0.9453 0.9480 0.9497 0.9477 0.9450 0.9417 0.9347 (10,10) 0.9197 0.9213 0.9230 0.9177 0.9133 0.9133 0.9097 (15,15) 0.9320 0.9323 0.9340 0.9333 0.9307 0.9277 0.9237 (2,4) (20,20) 0.9353 0.9373 0.9390 0.9387 0.9327 0.9310 0.9260 (25,25) 0.9287 0.9320 0.9333 0.9383 0.9333 0.9300 0.9263 (30,30) 0.9353 0.9380 0.9410 0.9397 0.9390 0.9393 0.9363 (35,35) 0.9453 0.9490 0.9490 0.9453 0.9433 0.9380 0.9360
4. Conclusions
In this paper, we have studied the multicomponent stress-strength reliability for generalized exponential distribution when both stress, strength variates follow the same population. Also, we have estimated asymptotic confidence interval for the multicomponent stress-strength reliability. The simulation results indicate that the average bias and average the MSE decrease as sample size increases for both situations of (s, k). Among the parameters the absolute bias and MSE decrease (increase) as α increases (β increases) in both the cases of(s, k). The length of the confidence interval also decreases as the sample size increases and the coverage
probability is close to the nominal value in all sets of parameters considered here.
Using real data, we illustrate the estimation process.
Recibido: abril de 2011 — Aceptado: diciembre de 2011
References
Awad, M. & Gharraf, K. (1986), ‘Estimation ofP(Y < X)in Burr case: A com- parative study’, Communications in Statistics-Simulation and Computation 15, 389–403.
Bhattacharyya, G. K. & Johnson, R. A. (1974), ‘Estimation of reliability in mul- ticomponent stress-strength model’, Journal of the American Statistical As- sociation69, 966–970.
David, H. A. (1981),Order Statistics, John Wiley and Sons, New York.
Downtown, F. (1973), ‘The estimation ofP(X > Y)in the normal case’,Techno- metrics15, 551–558.
Enis, P. & Geisser, S. (1971), ‘Estimation of the probability thatY < X’,Journal of the American Statistical Association66, 162–168.
Gupta, R. D. & Kundu, D. (1999), ‘Generalized exponential distributions’,Aus- tralian and New Zealand Journal of Statistics41, 173–188.
Gupta, R. D. & Kundu, D. (2001), ‘Generalized exponential distributions; different method of estimations’, Journal of Statistical Computation and Simulation 69, 315–338.
Gupta, R. D. & Kundu, D. (2002), ‘Generalized exponential distributions; statis- tical inferences’,Journal of Statistical Theory and Applications1, 101–118.
Gupta, R. D. & Kundu, D. (2003), ‘Discriminating between the Weibull and gener- alized exponential distributions’,Computational Statistics and Data Analysis 43, 179–196.
Kendall, M. G. & Stuart, A. (1979), The Advanced Theory of Statistics, Vol. 2, Charles Griffin and Company Limited, London.
Kundu, D. & Gupta, R. (2006), ‘Estimation ofP(Y < X)for Weibull distribution’, IEEE Transactions on Reliability55(2), 270–280.
Kundu, D. & Gupta, R. D. (2005), ‘Estimation ofP(Y < X) for the generalized exponential distribution’,Metrika61(3), 291–308.
Kundu, D. & Raqab, M. Z. (2009), ‘Estimation of R=P(Y < X) for three- parameter Weibull distribution’,Statistics and Probability Letters79, 1839–
1846.
Lawless, J. F. (1982), Statistical Models and Methods for Lifetime Data, John Wiley & Sons, New York.
Linhardt, H. & Zucchini, W. (1986),Model Selection, Wiley Eastern Limited, New York.
McCool, J. I. (1991), ‘Inference onP(Y < X)in the Weibull case’, Communica- tions in Statistics-Simulation and Computation20, 129–148.
Nandi, S. B. & Aich, A. B. (1994), ‘A note on estimation ofP(X > Y) for some distributions useful in life-testing’, IAPQR Transactions19(1), 35–44.
Pandey, M. & Uddin, B. M. (1985), Estimation of reliability in multicomponent stress-strength model following Burr distribution,in‘Proceedings of the First Asian congress on Quality and Reliability’, New Delhi, India, pp. 307–312.
Rao, C. R. (1973),Linear Statistical Inference and its Applications, Wiley Eastern Limited, India.
Rao, G. S. & Kantam, R. R. L. (2010), ‘Estimation of reliability in multicompo- nent stress-strength model: Log-logistic distribution’, Electronic Journal of Applied Statistical Analysis3(2), 75–84.
Raqab, M. Z. & Kundu, D. (2005), ‘Comparison of different estimators of P(Y < X) for a scaled Burr type X distribution’, Communications in Statistics-Simulation and Computation34(2), 465–483.
Raqab, M. Z., Modi, M. T. & Kundu, D. (2008), ‘Estimation of P(Y < X) for the 3-parameter generalized exponential distribution’, Communications in Statistics-Theory and Methods37(18), 2854–2864.
Surles, J. G. & Padgett, W. J. (1998), ‘Inference forP(Y < X)in the Burr type X model’,Communications in Statistics-Theory and Methods7, 225–238.
